Pericles S. Theocaris

Neural networks for computing in fracture mechanics. Methods and prospects of applications

Abstract A neural network is proposed and studied for the treatment of fracture mechanics problems. Both the cases of classical cracks and of cracks involving Coulombs friction or detachment (unilateral contact) interface conditions are considered. For the first case, the Hopfield model is appropriately modified, whereas for the second case, a neural model is proposed covering the case of inequalities. For this model, new results generalizing the results of Hopfield and Tank are obtained. Numerical applications illustrate the theory. Finally, the parameter identification problem for fractured bodies is formulated as a supervised learning problem.

Failure criteria for isotropic bodies revisited

Abstract Existing failure criteria for isotropic bodies are reconsidered in this paper and compared with modern versions taking into account either the influence of the strength differential effect or the influence of the internal dilation of the materials on yielding and therefore the contribution of the hydrostatic component of stress in failure. Modern criteria are expressed by quadric polynomials whose coefficients constitute convenient terms of the failure tensor of the material which for the isotropic body is defined by the respective failure stresses in simple tension and compression. Among the different expressions for the respective failure tensor polynomial of a material the paraboloid of revolution failure locus is the most convenient, since it fulfils the requirements of invariancy relative to any reference coordinate system, it is flexible and yields a unique solution for each loading path while it is unambiguously defined in the stress space. Furthermore, it is in conformity with basic physical laws and the extensive experience that the hydrostatic stress constitutes a safe loading path for the material. Experimental evidence with all varieties of isotropic materials corroborates the theory upon which the criterion is based. Finally, a failure criterion, based on void coalescence mechanisms inside the material, which also takes into consideration the influence of internal dilation of the material and therefore it depends on the hydrostatic component of stresses, is presented. This criterion is an improvement of the Gurson-McClintock criterion which permits a judicial determination of the coefficients of the respective quadric polynomial expressing it, since it belongs to the broad family of criteria based on energy principles.

The elliptic paraboloid failure criterion for cellular solids and brittle foams

SummaryCellular solids and brittle foams are increasingly finding application in constructions mainly as core materials for loaded sandwich structures where the loading of the structure generates multiaxial stress states on them. It has been established that the principal mechanism of deformation is based on the cell-wall bending and closed-cell as well as open-cell foams present similar stiffnesses. Therefore simple relations are found for their tensile, compressive and shear strengths and their elastic properties.It has been established in this paper that the modes of failure of such materials can be satisfactorily described by the elliptic paraboloid failure criterion for the general orthotropic body. Then, as soon as the yield or failure stresses in simple tension and compression are measured along the three principal stress directions of the material the failure locus is unequivocally defined and all the properties of the material under any complicated load can be accurately established. However, since these materials fail in the compression-compression-compression octant of the principal stress space by elastic buckling, the EPFS-criterion is cut-off by an ellipsoid surface, with intercepts along the principal axes the respective compressive failure stresses.The criterion thus established yields satisfactory results. It has been tested among others in a reticulated vitreous carbon foam as well as in an aluminium foam. The experimental results for these foams existing in the literature are fitting better with this universal criterion than any other considered, thus indicating the validity of the elliptic paraboloid failure criterion also for this type of materials.

Spectral decomposition of the compliance tensor for anisotropic plates

The spectral decomposition of compliance S is extended to the principal stress planes offering a possibility of characterization of the elastic properties of anisotropic media under plane-stress conditions. It is shown that the three eigenvalues of S, together with a “new” dimensionless parameter ωp, called the plane eigenangle, constitute the essential parameters for an invariant description of the elastic behaviour of anisotropic plates. Both the variational limits of the eigenangle ωp and the restrictive bounds to the values of the Poissons ratios imposed by thermodynamics are considered. Finally, it is shown that the plane eigenangle ωp may be employed as a monoparametric indication of the anisotropy of the material.

Invariant elastic constants and eigentensors of orthorhombic, tetragonal, hexagonal and cubic crystalline media

The purpose of this paper is to present a simple and direct way of determining the eigenvalues and eigentensors, as well as their orientations, for all crystals of the orthorhombic, tetragonal, hexagonal and cubic symmetries, a procedure based on the spectral decomposition of the compliance and stiffness fourth-rank tensors. First, both the eigenvalues and the idempotent fourth-rank tensors are derived for the orthorhombic and tetragonal-7 symmetries. The latter decompose, respectively, the second-rank symmetric tensor spaces of orthorhombic and tetragonal-7 media into orthogonal subspaces, consisting of the stress and strain eigentensors, and split the elastic potential into distinct noninteracting strain-energy parts. Accordingly, the spectrum of the compliance tensor of the tetragonal-6 symmetry is evaluated, by reduction of the eigenvalues and eigentensors of either the orthorhombic or tetragonal-7 symmetry. These results are, then, applied in turn to each of the hexagonal and cubic crystal systems. In each case, the eigenvalues, the idempotent tensors and the stress and strain eigentensors are easily derived as particular cases of the results obtained for the tetragonal-6 symmetry. Furthermore, it is noted that the positivity of the eigenvalues for each symmetry is equivalent to the positive definiteness of the elastic potential and, thus, necessary and sufficient conditions are acquired, in terms of the compliance-tensor components, characteristic of each symmetry.

Calculation of effective transverse elastic moduli of fiber-reinforced composites by numerical homogenization

Effective transverse elastic moduli for fiber-reinforced composites are calculated here by a numerical homogenization approach. The effects of fiber placement (staggering) and of weak-fiber and strong-matrix composites on the effective moduli, both of which are not very effectively treated by classical methods, are specifically investigated. Comparisons with classical, analytical approaches are included.

Weighing Failure Tensor Polynomial Criteria for Composites

Several interesting versions of the tensor polynomial failure criterion for composites were compared with the family of the elliptic paraboloid failure criteria (EPFS), with symmetry axes parallel to the hydrostatic axis. A critical appraisal of the most important theories was presented and a comparison was made with existing experi mental data in triaxial stress states. The superiority of the EPFS over any other criterion was established. Furthermore, the more flexible ellipsoid failure surface criterion, an ex tension of EPFS, may be used for special types of material such as foams, porous material and other materials that fail under internal buckling in general compression.

Failure modes of closed-cell polyurethane foams

The failure modes of closed-cell polyurethane foams were studied by applying the elliptic paraboloid failure surface criterion. A series of three polyurethane rigid foams (PUR-foams) were examined presenting different amounts of porosity from a highly porous material having a low density of 64 kg/m3 to a compact one with a density of 192 kg/m3. All these PUR-foams were of the same batch of material presenting a cell-wall density ϱ5=1200 kg/m3. Samples were tested in simple tension and compression along the three principal axes of anisotropy of the materials. It was shown that all three types of foams may be closely represented by transversely isotropic materials.The elliptic paraboloid failure surfaces (EPFS) for these three materials were defined from the six values of principal failure stresses in tension and compression. It was shown that the theoretically plotted paraboloid surfaces along all their principal-plane intersections were in good agreement with experiments.Since cellular materials collapse, either under elastic buckling in the compressive octant of the principal stress space, or under fast brittle fracture in the tensile octant, it was shown that the elliptic paraboloid failure surface is truncated by the intersection of the EPFS and an ellipsoid whose position and dimensions are interrelated with those of the EPFS. Again, experimental evidence with elastic buckling of foams corroborated the results of this theory.An important feature for the failure behavior of the foams was derived by this study according to which the foamed materials change mode of failure from a compression strong to a tension strong mode as their porosity is increased. In between they pass through a quasi-isotropic state.

Optimal material design in composites: An iterative approach based on homogenized cells

An iterative method is proposed for the optimal material design of structures. The case of optimal material design with composite materials, with materials of variable microstructures and some classes of optimal topology design problems are discussed. Homogenized elastic properties can be used based either on analytic results or on numerical homogenization techniques. Several numerical examples demonstrate the theory.

Multilevel optimal design of composite structures including materials with negative Poisson's ratio

Multilevel iterative optimal design procedures, horrowed from the theory of structural optimization by means of homogenization, are used in this paper for the optimal material design of composite material structures. The method is quite general and includes materials with appropriate microstructure, which may lead eventually to phenomenological, overall negative Poissons ratios. The benefits of optimal structural design gained by this approach, together with the first attempts to explain the taskoriented microstructure of natural structures, are investigated by means of numerical examples, and simulation of, among others, human bones.